Precoder structure for MIMO precoding

Abstract

The teachings herein present a method and apparatus that implement and use a factorized precoder structure that is advantageous in terms of performance and efficiency. In particular, the teachings presented herein disclose an underlying precoder structure that allows for certain codebook reuse across different transmission scenarios, including for transmission from a single Uniform Linear Array (ULA) of transmit antennas and transmission from cross-polarized subgroups of such antennas. According to this structure, an overall precoder is constructed from a conversion precoder and a tuning precoder. The conversion precoder includes antenna-subgroup precoders of size n_T/2, where n_T represents the number of overall antenna ports considered. Correspondingly, the tuning precoder controls the offset of beam phases between the antenna-subgroup precoders, allowing the conversion precoder to be used with cross-polarized arrays of n_T/2 antenna elements and with co-polarized arrays of n_T antenna elements.

Description

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where e_n is a noise/interference vector obtained as realizations of a random process and H_n is the complex channel. The precoder, W_NT×r, can be a wideband precoder, which is constant over frequency, or frequency selective.

Conventionally, the precoder matrix is often chosen to match the characteristics of the N_R×N_T MIMO channel matrix H, resulting in so-called channel dependent precoding. This is also commonly referred to as closed-loop precoding and essentially tries to focus the transmit energy into a subspace which is strong in the sense of conveying much of the transmitted energy to the targeted receiver. In addition, the precoder matrix also may be selected with the goal of orthogonalizing the channel, meaning that after proper linear equalization at a UE or other targeted receiver, the inter-layer interference is reduced.

According to the factorized precoder structure disclosed herein, the conversion precoders 32 are configured to have dimension N_T×k, where k is configurable and preferably is less than the number of transmit antenna ports N_T considered for precoding. In this regard k<N_T advantageously restricts the number of channel dimensions that must be accounted for in the tuning precoders 34. Correspondingly, the tuning precoders 34 are configured to have dimension k×r, where r is the transmission rank. This arrangement is shown below:
W_NT×r=W_NT×k^(c)W_k×r^(t),   (2)
where the conversion precoder 32, W_NT×k^(c), strives for capturing wideband/long-term properties of the channel such as correlation, while the tuning precoder 34, W_k×r^(t), targets frequency-selective/short-term properties of the channel.

The conversion precoder 32 exploits the correlation properties for focusing the tuning precoder 34 in “directions” where the propagation channel H on average is “strong.” Typically, this is accomplished by reducing the number of dimensions k covered by the tuning precoder 34. In other words, the conversion precoder 32 becomes a tall matrix with a reduced number of columns. Consequently, the number of rows k of the tuning precoder 34 is reduced as well. With such a reduced number of dimensions, the codebook used for storing the tuning precoders 34 can be made smaller, while still maintaining good performance.

In one arrangement already shown, the conversion precoders 32 are in one codebook 82 and the tuning precoders 34 are in another codebook 84. This arrangement exploits the fact that the conversion precoders 32 should have high spatial resolution and thus are advantageously implemented as a codebook 82 with many elements, while the codebook 84 for the tuning precoders 34 should be made small to keep the signaling overhead at a reasonable level.

To see how correlation properties are exploited and dimension reduction achieved, consider the case where the N_T different antennas 14 at the transceiver 10 are arranged into N_T/2 closely spaced cross-poles. Based on the polarization direction of the antenna subsets, the antennas in the closely spaced cross-pole setup can be divided into two groups, where each group is a closely spaced co-polarized Uniform Linear Array (ULA) with N_T/2 antennas. Closely spaced antennas often lead to high channel correlation and the correlation can in turn be exploited to maintain low signalling overhead. The channels corresponding to each such antenna group ULA are denoted H_/ and H_\, respectively.

For convenience in notation, the following equations drop the subscripts indicating the dimensions of the matrices as well as the subscript n. Assume that each conversion precoder 32 has a block diagonal structure,

$\begin{array}{cc}{W}^{\left(c\right)}=\left[\begin{array}{cc}{\tilde{W}}^{\left(c\right)}& 0\\ 0& {\tilde{W}}^{\left(c\right)}\end{array}\right].& \left(3\right)\end{array}$
The product of the MIMO channel H and the overall precoder 36 can then be written as:

$\begin{array}{cc}\begin{array}{c}\mathrm{HW}=\left[\begin{array}{cc}{H}_{/}& H_{\backslash }\end{array}\right]W^{\left(c\right)}{W}^{\left(t\right)}\\ =\left[\begin{array}{cc}{H}_{/}& H_{\backslash }\end{array}\right]\left[\begin{array}{cc}{\tilde{W}}^{\left(c\right)}& 0\\ 0& {\tilde{W}}^{\left(c\right)}\end{array}\right]W^{\left(t\right)}\\ =\left[\begin{array}{cc}{H}_{/}{\tilde{W}}^{\left(c\right)}& H_{\backslash }\end{array}{\tilde{W}}^{\left(c\right)}\right]W^{\left(t\right)}\\ =H_{\mathrm{eff}}{W}^{\left(t\right)}\end{array}& \left(4\right)\end{array}$
As seen, the matrix {tilde over (W)}^(c) separately precodes each antenna group ULA, thereby forming a smaller and improved effective channel H_eff. As such, the blocks within W^(c) are referred to as antenna subgroup precoders 38. If {tilde over (W)}^(c) corresponds to a beamforming vector, the effective channel would reduce to having only two virtual antennas, which reduces the needed size of the codebook(s) 30 used for the second tuning precoder matrix W^(t) when tracking the instantaneous channel properties. In this case, instantaneous channel properties are to a large extent dependent upon the relative phase relation between the two orthogonal polarizations.

It is also helpful for a fuller understanding of this disclosure to consider the theory regarding a “grid of beams,” along with Discrete Fourier Transform (DFT) based precoding. DFT based precoder vectors for N_T transmit antennas can be written in the form:

$\begin{array}{cc}{w}_n^{\left(N_T,Q\right)}={\left[\begin{array}{cccc}{w}_{1,n}^{\left(N_T,Q\right)}& w_{2,n}^{\left(N_T,Q\right)}& \dots & w_{N_T,n}^{\left(N_T,Q\right)}\end{array}\right]}^T{w}_{m,n}^{\left(N_T,Q\right)}=\exp \left(j\frac{2\pi }{N_{T}Q}mn\right),m=0,\dots ,N_T-1,n=0,\dots ,{\mathrm{QN}}_T-1,& \left(5\right)\end{array}$
where w_m,n^(NT,Q) is the phase of the m:th antenna, n is the precoder vector index (i.e., which beam out of the QN_T beams) and Q is the oversampling factor. As seen, the phase increases with the same amount from one antenna port to another, i.e., linearly growing phase with respect to the antenna port index m. This is in fact a characteristic of DFT-based precoding. Thus DFT based precoder vectors may include additional phase shifts on top of those shown in the above expression as long as the overall phase shift is increasing linearly with m.

For good performance, it is important that the array gain function of two consecutive transmit beams overlaps in the angular domain, so that the gain does not drop too much when going from one beam to another. This requires an oversampling factor of at least Q=2. Thus for N_T antennas, at least 2N_T beams are needed.

An alternative parameterization of the above DFT based precoder vectors is:

$\begin{array}{cc}{w}_{l,q}^{\left(N_T,Q\right)}={\left[\begin{array}{cccc}{w}_{1,\mathrm{Ql}+q}^{\left(N_T,Q\right)}& w_{2,\mathrm{Ql}+q}^{\left(N_T,Q\right)}& \dots & w_{N_T,\mathrm{Ql}+q}^{\left(N_T,Q\right)}\end{array}\right]}^T{w}_{m,\mathrm{Ql}+q}^{\left(N_T,Q\right)}=\exp \left(j\frac{2\pi }{N_T}m\left(l+\frac{q}{Q}\right)\right),& \left(6\right)\end{array}$
for m=0, . . . , N_T−1, l=0, . . . , N_T−1, q=0,1, . . . , Q−1, and where l and q together determine the precoder vector index via the relation n=Ql+q. This parameterization also highlights that there are Q groups of beams, where the beams within each group are orthogonal to each other. The q:th group can be represented by the generator matrix:

$\begin{array}{cc}{G}_q^{\left(N_T\right)}=[\begin{array}{cccc}{w}_{0,q}^{\left(N_T,Q\right)}& w_{l,q}^{\left(N_T,Q\right)}& \dots & w_{N_T-1,q}^{\left(N_T,Q\right)}].\end{array}& \left(7\right)\end{array}$
By insuring that only precoder vectors from the same generator matrix are being used together as columns in the same precoder, it is straightforward to form sets of precoder vectors for use in so-called unitary precoding where the columns within a precoder matrix should form an orthonormal set.

Further, to maximize the performance of DFT based precoding, it is useful to center the grid of beams symmetrically around the broad size of the array. Such a rotation of the beams can be done by multiplying from the left the above DFT vectors w_n^(NT,Q) with a diagonal matrix W_rot having elements:

$\begin{array}{cc}{\left[W_{\mathrm{rot}}\right]}_{\mathrm{mm}}=\exp \left(j\frac{\pi }{{\mathrm{QN}}_T}m\right).& \left(8\right)\end{array}$
The rotation can either be included in the precoder codebook or alternatively can be carried out as a separate step where all signals are rotated in the same manner and the rotation can thus be absorbed into the channel from the perspective of the receiver (transparent to the receiver). For the remainder of DFT-precoding discussion herein, it is tacitly assumed that rotation may or may not have been carried out as part of DFT-based precoding.

One aspect of the above-described factorized precoder structure relates to lowering the overhead associated with signaling the conversion and tuning precoders 32 and 34, based on signaling them with different frequency and/or time granularity. The use of a block diagonal conversion precoder 32 is specifically optimized for the case of a transmit antenna array comprising closely spaced cross-poles, but other antenna arrangements exist as well. In particular, efficient performance with a ULA of closely spaced co-poles should also be achieved using the same conversion precoders 32. The precoder structures disclosed herein advantageously provide for use of the same conversion precoder structure, irrespective of whether the transceiver 10 uses its antennas as a ULA of N_T closely-spaced co-poles, or as two subsets cross-poles, each subset having N_T/2 antenna elements.

In particular, in one or more embodiments, the conversion precoders 32 comprise DFT-based precoders which are suitable for the two N_T/2 element antenna group ULAs in a closely spaced cross-pole setup, while still providing for their re-use in forming the needed number of DFT based size N_T precoders for an N_T element ULA. Moreover, one or more embodiments disclosed herein provide a structure for the conversion precoder that allows re-using existing codebooks with DFT based precoders and extending their spatial resolution.

In any case, an example embodiment illustrates re-using DFT based precoder elements for an antenna group ULA in a closely spaced cross-pole and also in creating a grid of beams with sufficient overlap for a ULA of twice the number of elements compared with the antenna group ULA. In other words, the conversion precoders 32 can be designed for use with the multiple antennas 14 of the transceiver 10, regardless of whether those antennas 14 are configured and operated as an overall ULA of N_T antennas, or as two cross-polarized ULA sub-groups, each having N_T/2 antennas.

Consider again the block diagonal factorized precoder design given as:

$\begin{array}{cc}W=W^{\left(c\right)}{W}^{\left(t\right)}=\left[\begin{array}{cc}{\tilde{W}}^{\left(c\right)}& 0\\ 0& {\tilde{W}}^{\left(c\right)}\end{array}\right]W^{\left(t\right)},& \left(9\right)\end{array}$
and note that in order to tailor the transmission to ±45 degrees cross-poles, the structure of a conversion precoder 32 can be modified by means of a multiplication from the left with a matrix:

$\begin{array}{cc}\left[\begin{array}{cc}I& I{e}^{j\varphi }\\ I& -I{e}^{j\varphi }\end{array}\right],& \left(10\right)\end{array}$
which, for ϕ=0, rotates the polarizations 45 degrees to align with horizontal and vertical polarization. Other values of ϕ may be used to achieve various forms of circular polarization.

For an N_T element ULA, the overall precoder 36 for rank 1 is to be an N_T×1 vector as:

$\begin{array}{cc}W=w_n^{\left(N_T,Q\right)}={\left[\begin{array}{cccc}{w}_{1,n}^{\left(N_T,Q\right)}& w_{2,n}^{\left(N_T,Q\right)}& \dots & w_{N_T,n}^{\left(N_T,Q\right)}\end{array}\right]}^T.& \left(11\right)\end{array}$
For antennas m=0, 1, . . . , N_T/2−1,

$\begin{array}{cc}{w}_{m,n}^{\left(N_T,Q\right)}=\exp \left(j\frac{2\pi }{N_{T}Q}\mathrm{mn}\right)=\exp \left(j\frac{2\pi }{\frac{N_T}{2}\left(2Q\right)}\mathrm{mn}\right)=w_{m,n}^{\left(N_T/2,2Q\right)},n=0,\dots ,{\mathrm{QN}}_T-1,& \left(12\right)\end{array}$
while for the remaining antennas m=N_T/2+m′, m′=0, 1, . . . , N_T/2−1,

$\begin{array}{cc}{w}_{N_T/2+m^{\prime }n}^{\left(N_T,Q\right)}=\exp \left(j\frac{2\pi }{N_{T}Q}\left(N_T/2+m^{\prime }\right)n\right)& \left(13\right)\end{array}$ $=\exp \left(j\frac{2\pi }{\frac{N_T}{2}\left(2Q\right)}{m}^{\prime }n\right)\exp \left(j\frac{\pi }{Q}n\right)$ $=w_{m^{\prime },n}^{\left(N_T/2,2Q\right)}\exp \left(j\frac{\pi }{Q}n\right)$ $=w_{m^{\prime },n}^{\left(N_T/2,2Q\right)}\alpha ,$ $n=0,\dots ,{\mathrm{QN}}_T-1.$
Here,

$\alpha \in \left\{\exp \left(j\frac{\pi }{Q}n\right):n=0,1,\dots ,2Q-1\right\}.$

Any N_T element DFT overall precoder 36 can thus be written as:

$\begin{array}{cc}{w}_n^{\left(N_T,Q\right)}=& \left(14\right)\end{array}$ ${\left[\begin{array}{cccccc}{w}_{0,n}^{\left(N_T,Q\right)}& w_{1,n}^{\left(N_T,Q\right)}& \dots & w_{N_T-1,n}^{\left(N_T,Q\right)}& w_{0,n}^{\left(N_T,Q\right)}\alpha & w_{1,n}^{\left(N_T,Q\right)}\alpha \dots w_{N_T-1,n}^{\left(N_T,Q\right)}\alpha \end{array}\right]}^T=$ $\left[\begin{array}{c}{w}_n^{\left(N_T/2,2Q\right)}\\ w_n^{\left(N_T/2,2Q\right)}\alpha \end{array}\right]=\left[\begin{array}{cc}{w}_n^{\left(N_T/2,2Q\right)}& 0\\ 0& w_n^{\left(N_T/2,2Q\right)}\end{array}\right]\left[\begin{array}{c}1\\ \alpha \end{array}\right].$
One sees in the above arrangement that w_n^(NT,Q) may be regarded as an example of an overall precoder 36 formed from a conversion precoder 32 given as:

$\left[\begin{array}{cc}{w}_n^{\left(N_T/2,2Q\right)}& 0\\ 0& w_n^{\left(N_T/2,2Q\right)}\end{array}\right],$
and a tuning precoder 34 given as

$\left[\begin{array}{c}1\\ \alpha \end{array}\right].$
Note further that each block, w_n^(NT/2,2Q), of the conversion precoder 32 represents one of the antenna-subgroup precoders 38 included in the conversion precoder 32, and note that the tuning precoders 34 are determined as:

$\begin{array}{cc}\left\{\left[\begin{array}{c}1\\ \exp \left(j\frac{\pi }{Q}n\right)\end{array}\right]:n=0,1,\dots ,2Q-1\right\}.& \left(15\right)\end{array}$

The above arrangement suits the closely spaced cross-polarized antenna array perfectly because size N_T/2 DFT-based antenna-subgroup precoders 38 are now applied on each antenna group ULA and the tuning precoder 34 provides 2Q different relative phase shifts between the two orthogonal polarizations. It is also seen how the N_T/2 element antenna-subgroup precoders 38 are reused for constructing the N_T element overall precoder 36. Of further note, the oversampling factor Q is twice as large in the cross-polarized case as it is for the co-polarized case, but those elements are not wasted because they help to increase the spatial resolution of the grid of beams precoders even further. This characteristic is particularly useful in MU-MIMO applications where good performance relies on the ability to very precisely form beams towards the UE of interest and nulls towards the other co-scheduled UEs.

For example, take a special case of N_T=8 transmit antennas—i.e., assume that the transceiver 10 of FIG. 1 includes eight antennas 14, for use in precoded MIMO transmissions, and assume that Q=2 for the closely spaced ULA. One sees that the overall precoder 36 is built up as:

$\begin{array}{cc}\begin{array}{c}\begin{array}{c}{w}_n^{\left(8,2\right)}=\left[\begin{array}{c}{w}_n^{\left(N_T/2,2Q\right)}\\ w_n^{\left(N_T/2,2Q\right)}\alpha \end{array}\right]\\ =\left[\begin{array}{cc}{w}_n^{\left(4,4\right)}& 0\\ 0& w_n^{\left(4,4\right)}\end{array}\right]\left[\begin{array}{c}1\\ \exp \left(j\frac{\pi }{2}{n}^{\prime }\right)\end{array}\right],\end{array}\\ n=0,\dots ,2N_T-1,n^{\prime }=0,1,2,3.\end{array}& \left(16\right)\end{array}$
The codebook entries for the tuning precoders 34 can then be chosen from the rank 1,2 Tx codebook in LTE and hence that codebook can be re-used in the teachings disclosed herein. The codebook for the conversion precoders 32 contains elements constructed from four DFT based generator matrices as in Eq. (7). The codebook(s) 30 can contain other elements in addition to the DFT based ones being described here. Broadly, the principle of constructing N element DFT-based overall precoders 36 out of smaller, N/2 element DFT-based antenna-subgroup precoders 38 can be used in general to add efficient closely spaced ULA and cross-pole support to a range of codebook-based precoding schemes. As a further advantage, the disclosed precoder structure can be used even if the antenna setups differ from what is being discussed here.

Further, note that DFT-based overall precoders 36 can be used for higher transmission ranks than one. One way to accomplish this is to pick the conversion precoders 32 as column subsets of DFT-based generator matrices, such as shown in Eq. (7). The tuning precoders 34 can be extended with additional columns as well, to match the desired value of the transmission rank. For transmission rank 2, a tuning precoder 34 can be structured as:

$\begin{array}{cc}{W}^{\left(t\right)}=\left[\begin{array}{cc}1& 1\\ \alpha & -\alpha \end{array}\right],\alpha \in \left\{\exp \left(j\frac{\pi }{Q}n\right):n=0,1,\dots ,2Q-1\right\}.& \left(17\right)\end{array}$

It is sometimes beneficial to re-use existing codebooks in the design of new codebooks. However, one associated problem is that existing codebooks may not contain all the needed DFT precoder vectors to provide at least Q=2 times oversampling of the grid of beams. Assuming for example that one has an existing codebook for N_T/2 antennas with DFT precoders providing Q=Q_e in oversampling factor and that the target oversampling factor for the N_T/2 element antenna group ULA is Q=Q_t. The spatial resolution of the existing codebook can then be improved to the target oversampling factor in factorized precoder design as:

$\begin{array}{cc}w=\left[\begin{array}{cc}{\Lambda }_{\tilde{q}}{w}_n^{\left(N_T/2,Q_e\right)}& 0\\ 0& {\Lambda }_{\tilde{q}}{w}_n^{\left(N_T/2,Q_e\right)}\end{array}\right]\left[\begin{array}{c}1\\ \alpha \end{array}\right],n=0,\dots ,Q_e{N}_T-1,\tilde{q}=0,1,\dots ,Q_t/Q_e-1{\Lambda }_{\tilde{q}}=\mathrm{diag}\left(1,\exp \left(j\frac{2\pi }{\frac{N_T}{2}}\frac{\tilde{q}}{Q_t}1\right),\exp \left(j\frac{2\pi }{\frac{N_T}{2}}\frac{\tilde{q}}{Q_t}2\right),\dots ,\exp \left(j\frac{2\pi }{\frac{N_T}{2}}\frac{\tilde{q}}{Q_t}\left(N_T/2-1\right)\right)\right).& \left(18\right)\end{array}$
Here, the w_n^(NT/2,Qe) could be elements in the existing LTE 4 Tx House Holder codebook, which contains 8 DFT based precoders (using an oversampling factor of Q=2 so that there is some overlap among the beams spanning four antennas) for rank 1. When the transmission rank is higher than one, the block diagonal structure can be maintained and the structure thus generalizes to:

$\begin{array}{cc}W=\left[\begin{array}{cc}{\Lambda }_{\tilde{q}}{\tilde{W}}^{\left(c\right)}& 0\\ 0& {\Lambda }_{\tilde{q}}{\tilde{W}}^{\left(c\right)}\end{array}\right]W^{\left(t\right)},& \left(19\right)\end{array}$
where W is now an N_T×r matrix, {tilde over (W)}^(c) is a matrix with at least one column equal to a DFT based antenna-subgroup precoder w_n^(NT/2,Qe), and the tuning precoder W^(t) has r columns.

To see that that the spatial resolution can be improved by multiplying an antenna-subgroup precoder 38 with a diagonal matrix as described above, consider the alternative parameterization of DFT precoders in Eq. (6),

$\begin{array}{cc}{w}_{m,Q_{t}l+q}^{\left(N_T,Q_t\right)}=\exp \left(j\frac{2\pi }{N_T}m\left(l+\frac{q}{Q_t}\right)\right),m=0,\dots ,N_T-1,l=0,\dots ,N_T-1,q=0,\dots ,Q_t-1,& \left(20\right)\end{array}$
l=0, . . . , N_T−1, q=0, . . . , Q_t−1,

and let:

$\begin{array}{cc}q=\frac{Q_t}{Q_e}{q}^{\prime }+\tilde{q},q^{\prime }=0,\dots ,Q_e-1,\tilde{q}=0,\dots ,\frac{Q_t}{Q_e}-1,& \left(21\right)\end{array}$
to arrive at:

$\begin{array}{cc}\begin{array}{c}{w}_{m,Q_{t}l+\frac{Q_t}{Q_e}{q}^{\prime }+\tilde{q}}^{\left(N_T,Q_t\right)}=\exp \left(j\frac{2\pi }{N_T}m\left(l+\frac{1}{Q_t}\left(\frac{Q_t}{Q_e}{q}^{\prime }+\tilde{q}\right)\right)\right)\\ =\exp \left(j\frac{2\pi }{N_T}m\left(l+\frac{q^{\prime }}{Q_e}\right)\right)\exp \left(j\frac{2\pi }{N_T}m\frac{\tilde{q}}{Q_t}\right)\\ =w_{m,Q_{e}l+q^{\prime }}^{\left(N_T,Q_e\right)}\exp \left(j\frac{2\pi }{N_T}m\frac{\tilde{q}}{Q_t}\right)\end{array}\mathrm{for}m=0,\dots ,N_T-1,l=0,\dots ,N_T-1,q^{\prime }=0,\dots ,Q_e-1,\tilde{q}=0,\dots ,\frac{Q_t}{Q_e}-1.& \left(22\right)\end{array}$

The above formulations demonstrate an advantageous aspect of the teachings presented herein. Namely, a codebook containing DFT precoders with oversampling factor Q_e can be used for creating a higher resolution DFT codebook by multiplying the m:th element with exp

$\left(j\frac{2\pi }{N_T}m\frac{\tilde{q}}{Q_t}\right)$
and hence proving that the diagonal transformation given by A_{{tilde over (q)}} indeed works as intended.

Another issue to take into account when designing precoders is to ensure an efficient use of the power amplifiers (PAs), e.g., the PAs in the transmitters 18 used for multi-antenna transmission from the transceiver 10. Usually, power cannot be borrowed across antennas because there is a separate PA for each antenna. Hence, for maximum use of the PA resources, it is important that the same amount of power is transmitted from each antenna. In other words, an overall precoder matrix W for precoding from the transmit antennas should fulfill
[WW*]_mm=κ, ∀m.   (23)

Thus, it is beneficial from a PA utilization point of view to enforce this constraint when designing precoder codebooks. Full power utilization is also ensured by the so-called constant modulus property, which means that all scalar elements in a precoder have the same norm (modulus). It is easily verified that a constant modulus precoder also fulfills the full PA utilization constraint in Eq. (23). Hence, the constant modulus property constitutes a sufficient but not necessary condition for full PA utilization.

With the beneficial aspect of full PA utilization in mind, another aspect of the teachings presented herein relates to providing precoders that yield full PA utilization. In particular, one or more embodiments proposed herein solve the problems associated with full PA utilization and satisfaction of the rank nested property, in the context of a factorized precoder design. By using a so-called double block diagonal tuning precoder 34 combined with a block diagonal conversion precoder 32, full PA utilization is guaranteed and rank override exploiting the nested property is also possible for the overall precoder formed as the combination of a conversion precoder 32 and a tuning precoder 34 having the properties and structure disclosed herein.

A first step in designing efficient factorized precoder codebooks while achieving full PA utilization and fulfilling rank nested property is to make the conversion precoders block diagonal as shown in Eq. (3), for example. In a particular case, the number of columns k of a conversion precoder is made equal to 2┌r/2┐, where ┌·┐ denotes the ceil function. This structure is achieved by adding two new columns contributing equally much to each polarization for every other rank. In other words, the conversion precoder 32 at issue here can be denoted as W^(c) and written in the form:

$\begin{array}{cc}{w}^{\left(c\right)}=\left[\begin{array}{cc}{\tilde{w}}^{\left(c\right)}& 0\\ 0& {\tilde{w}}^{\left(c\right)}\end{array}\right]=\left[\begin{array}{cccccccc}{\tilde{w}}_1^{\left(c\right)}& {\tilde{w}}_2^{\left(c\right)}& \dots & {\tilde{w}}_{\lceil r/2\rceil }^{\left(c\right)}& 0& 0& \dots & 0\\ 0& 0& \dots & 0& {\tilde{w}}_1^{\left(c\right)}& {\tilde{w}}_2^{\left(c\right)}& \dots & {\tilde{w}}_{\lceil r/2\rceil }^{\left(c\right)}\end{array}\right],& \left(24\right)\end{array}$
where {tilde over (w)}_l^(c) is an N_T/2×1 vector.

Extending the conversion dimension in this manner helps keep the number of dimensions small and in addition serves to make sure that both polarizations are excited equally much. It is beneficial if the conversion precoder, denoted here as {tilde over (W)}^(c), is also made to obey a generalized rank nested property in that there is freedom to choose {tilde over (W)}^(c) with L columns as an arbitrary column subset of each possible {tilde over (W)}^(c) with L+1 columns. An alternative is to have the possibility to signal the column ordering used in {tilde over (W)}^(c). Flexibility in the choice of columns for {tilde over (W)}^(c) for the different ranks is beneficial so as to still be able to transmit into the strongest subspace of the channel even when rank override using a column subset is performed.

Further, as regards to ensuring full PA utilization, e.g., at the transceiver 10, the tuning precoders 34, which are denoted as W^(t), are in one or more embodiments constructed as follows: (a) the conversion vector {tilde over (w)}_n^(c) is made constant modulus; and (b) a column in the tuning precoder has exactly two non-zero elements with constant modulus. If the m:th element is non-zero, so is element m+┌r/2┐. Hence for rank r=4, the columns in the tuning precoder 34 are of the following form:

$\begin{array}{cc}\left[\begin{array}{c}x\\ 0\\ x\\ 0\end{array}\right],\left[\begin{array}{c}0\\ x\\ 0\\ x\end{array}\right],& \left(25\right)\end{array}$
where x denotes an arbitrary non-zero value which is not necessarily the same from one x to another. Because there are two non-zero elements in a column, two orthogonal columns with the same positions of the non-zero elements can be added before columns with other non-zero positions are considered. Such pairwise orthogonal columns with constant modulus property can be parameterized as:

$\begin{array}{cc}\left[\begin{array}{c}1\\ 0\\ e^{j\varphi }\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 0\\ -e^{j\varphi }\\ 0\end{array}\right].& \left(26\right)\end{array}$
Rank nested property for the overall precoder is upheld when increasing the rank by one by ensuring that columns for previous ranks excite the same columns of the conversion precoder also for the higher rank. Combining this with Eq. (25) and the mentioned pairwise orthogonal property of the columns leads to a double block diagonal structure of the tuning precoder taking the form:

$\begin{array}{cc}W=\left[\text{⁠}\begin{array}{cccccccc}{\tilde{w}}_1^{\left(c\right)}& {\tilde{w}}_2^{\left(c\right)}& \dots & {\tilde{w}}_{\lceil r/2\rceil }^{\left(c\right)}& 0& 0& \dots & 0\\ 0& 0& \dots & 0& {\tilde{w}}_1^{\left(c\right)}& {\tilde{w}}_2^{\left(c\right)}& \dots & {\tilde{w}}_{\lceil r/2\rceil }^{\left(c\right)}\end{array}\right]& \left(27\right)\end{array}$ $\left[\begin{array}{ccccccc}x& x& 0& 0& \dots & & \\ 0& 0& x& x& & & \\ ⋮& & & & \ddots & & \\ & & & & & & \\ x& x& 0& 0& \dots & & \\ 0& 0& x& x& & & \\ ⋮& & & & \ddots & & \\ & & & & & & \end{array}\right].$
Using the pairwise orthogonality property in Eq. (26), and representing the structure for the overall precoder 36, denoted as W, as W=W^(c)W^(t), the precoder structure can be further specialized into:

$\begin{array}{cc}W=\left[\begin{array}{cccccccc}{\tilde{w}}_1^{\left(c\right)}& {\tilde{w}}_2^{\left(c\right)}& \dots & {\tilde{w}}_{\lceil r/2\rceil }^{\left(c\right)}& 0& 0& \dots & 0\\ 0& 0& \dots & 0& {\tilde{w}}_1^{\left(c\right)}& {\tilde{w}}_2^{\left(c\right)}& \dots & {\tilde{w}}_{\lceil r/2\rceil }^{\left(c\right)}\end{array}\right]& \left(28\right)\end{array}$ $\left[\text{⁠}\begin{array}{cccccc}1& 1& 0& 0& \dots & \\ 0& 0& 1& 1& & \\ ⋮& & & & \ddots & \\ & & & & & \\ e^{j{\varphi }_1}& -e^{j{\varphi }_1}& 0& 0& \dots & \\ 0& 0& e^{j{\varphi }_2}& -e^{j{\varphi }_2}& & \\ ⋮& & & & \ddots & \\ & & & & & \end{array}\right].$
Note that the double block diagonal structure for the tuning precoder can be described in different ways depending on the ordering of the columns used for storing the conversion precoders W^(c) as entries in the codebook 30. It is possible to equivalently make the tuning precoders W^(t) block diagonal by writing:

$\begin{array}{cc}W=\left[\begin{array}{cccccccc}{\tilde{w}}_1^{\left(c\right)}& 0& {\tilde{w}}_2^{\left(c\right)}& 0& \dots & \dots & {\tilde{w}}_{\lceil r/2\rceil }^{\left(c\right)}& 0\\ 0& {\tilde{w}}_1^{\left(c\right)}& 0& {\tilde{w}}_2^{\left(c\right)}& \dots & \dots & 0& {\tilde{w}}_{\lceil r/2\rceil }^{\left(c\right)}\end{array}\right]& \left(29\right)\end{array}$ $\left[\begin{array}{cccccccc}x& x& 0& 0& \dots & & 0& 0\\ x& x& 0& 0& & & & ⋮\\ 0& 0& x& x& \ddots & & & \\ ⋮& & x& x& & & & \\ & & 0& 0& \ddots & & & \\ & & ⋮& & & & 0& 0\\ & & & & \ddots & & x& x\\ 0& 0& 0& 0& \dots & 0& x& x\end{array}\right].$
Re-orderings similar to these do not affect the overall precoder W and are thus considered equivalent and assumed to be covered under the terms “block diagonal conversion precoder and double block diagonal tuning precoder.” It is also interesting to note that if the requirements on the orthogonality constraint and full PA utilization are relaxed, the design for rank nested property can be summarized with the following structure for the tuning precoders 34:

$\begin{array}{cc}\left[\begin{array}{cccccccc}x& x& x& x& x& x& & \\ 0& 0& x& x& x& x& & \\ ⋮& & & & x& x& \ddots & \\ & & & & & & & \\ x& x& x& x& x& x& & \\ 0& 0& x& x& x& x& & \\ ⋮& & & & x& x& \ddots & \\ & & & & & & & \end{array}\right].& \left(30\right)\end{array}$

Further, it is worth mentioning that rank nested property can be useful when applied separately to the conversion precoders 32 and the tuning precoders 34. Even applying it only to the tuning precoders 34 can help save computational complexity, because precoder calculations across ranks can be re-used as long as the selected conversion precoder W^(c) remains fixed.

As an illustrative example for eight transmit antennas 14 at the transceiver 10, assume that

$\mathrm{Rank}r=1:$ $\begin{array}{cc}W=\left[\begin{array}{cc}{w}_1^{\left(1\right)}& \\ & w_1^{\left(1\right)}\end{array}\right]\left[\begin{array}{c}1\\ e^{j{\phi }_k}\end{array}\right]& \left(31\right)\end{array}$ $\mathrm{Rank}r=2:$ $\begin{array}{cc}W=\left[\begin{array}{cc}{w}_1^{\left(1\right)}& \\ & w_1^{\left(1\right)}\end{array}\right]\left[\begin{array}{cc}1& 1\\ e^{j{\phi }_k}& -e^{j{\phi }_k}\end{array}\right]& \left(32\right)\end{array}$ $\mathrm{Rank}r=3:$ $\begin{array}{cc}W=\left[\begin{array}{cccc}{w}_1^{\left(1\right)}& w_2^{\left(1\right)}& & \\ & & w_1^{\left(1\right)}& w_2^{\left(1\right)}\end{array}\right]\left[\begin{array}{ccc}1& 1& 0\\ 0& 0& 1\\ e^{j{\phi }_k}& e^{j{\phi }_k}& 0\\ 0& 0& e^{j{\phi }_l}\end{array}\right]& \left(33\right)\end{array}$ $\mathrm{Rank}r=4:$ $\begin{array}{cc}W=\left[\begin{array}{cccc}{w}_1^{\left(1\right)}& w_2^{\left(1\right)}& & \\ & & w_1^{\left(1\right)}& w_2^{\left(1\right)}\end{array}\right]\left[\begin{array}{cccc}1& 1& 0& 0\\ 0& 0& 1& 1\\ e^{j{\phi }_k}& -e^{j{\phi }_k}& 0& 0\\ 0& 0& e^{j{\phi }_l}& -e^{j{\phi }_l}\end{array}\right]& \left(34\right)\end{array}$ $\mathrm{Rank}r=5:$ $\begin{array}{cc}W=\left[\begin{array}{cccccc}{w}_1^{\left(1\right)}& w_2^{\left(1\right)}& w_3^{\left(1\right)}& & & \\ & & & w_1^{\left(1\right)}& w_2^{\left(1\right)}& w_3^{\left(1\right)}\end{array}\right]& \left(35\right)\end{array}$ $\left[\begin{array}{ccccc}1& 1& 0& 0& 0\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 0& 1\\ e^{j{\phi }_k}& -e^{j{\phi }_k}& 0& 0& 0\\ 0& 0& e^{j{\phi }_l}& -e^{j{\phi }_l}& 0\\ 0& 0& 0& 0& e^{j{\phi }_m}\end{array}\right]$ $\mathrm{Rank}r=6:$ $\begin{array}{cc}W=\left[\begin{array}{cccccc}{w}_1^{\left(1\right)}& w_2^{\left(1\right)}& w_3^{\left(1\right)}& & & \\ & & & w_1^{\left(1\right)}& w_2^{\left(1\right)}& w_3^{\left(1\right)}\end{array}\right]& \left(36\right)\end{array}$ $\left[\begin{array}{cccccc}1& 1& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 1& 1\\ e^{j{\phi }_k}& -e^{j{\phi }_k}& 0& 0& 0& 0\\ 0& 0& e^{j{\phi }_l}& -e^{j{\phi }_l}& 0& 0\\ 0& 0& 0& 0& e^{j{\phi }_m}& -e^{j{\phi }_m}\end{array}\right]$ $\mathrm{Rank}r=7:$ $\begin{array}{cc}W=\left[\begin{array}{ccccccc}{w}_1^{\left(1\right)}& w_2^{\left(1\right)}& w_3^{\left(1\right)}& w_4^{\left(1\right)}& & & \\ & & & & w_1^{\left(1\right)}& w_2^{\left(1\right)}& w_3^{\left(1\right)}{w}_4^{\left(1\right)}\end{array}\right]& \left(37\right)\end{array}$ $\left[\begin{array}{ccccccc}1& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\\ e^{j{\phi }_k}& -e^{j{\phi }_k}& 0& 0& 0& 0& 0\\ 0& 0& e^{j{\phi }_l}& -e^{j{\phi }_l}& 0& 0& 0\\ 0& 0& 0& 0& e^{j{\phi }_m}& -e^{j{\phi }_m}& 0\\ 0& 0& 0& 0& 0& 0& e^{j{\phi }_n}\end{array}\right]$ $\mathrm{Rank}r=8:$ $\begin{array}{cc}W=\left[\begin{array}{ccccccc}{w}_1^{\left(1\right)}& w_2^{\left(1\right)}& w_3^{\left(1\right)}& w_4^{\left(1\right)}& & & \\ & & & & w_1^{\left(1\right)}& w_2^{\left(1\right)}& w_3^{\left(1\right)}{w}_4^{\left(1\right)}\end{array}\right]& \left(38\right)\end{array}$ $\left[\begin{array}{cccccccc}1& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 1\\ e^{j{\phi }_k}& -e^{j{\phi }_k}& 0& 0& 0& 0& 0& 0\\ 0& 0& e^{j{\phi }_l}& -e^{j{\phi }_l}& 0& 0& 0& 0\\ 0& 0& 0& 0& e^{j{\phi }_m}& -e^{j{\phi }_m}& 0& 0\\ 0& 0& 0& 0& 0& 0& e^{j{\phi }_n}& -e^{j{\phi }_n}\end{array}\right]$
The four Tx case follows in a similar manner.

With the above in mind, the following structure and provisions are proposed herein, for one or more embodiments that provide for full PA utilization:

• • 1. The overall precoder 36 can be factorized into a conversion precoder 32 and a tuning precoder 34.
    • a. the conversion precoder 32 is block diagonal
    • b. the tuning precoder 34 has the properties:
      • i. all non-zero elements are constant modulus
      • ii. every column has exactly two non-zero elements
      • iii. every row has exactly two non-zero elements
  • 2. Two columns in the tuning precoder 34 either have non-zero elements in the same two rows or do not have any non-zero elements in the same rows.
  • 3. Two columns in the tuning precoder 34 having non-zero elements in the same two rows are orthogonal to each other.
  • 4. The conversion precoder 32 has 2┌k/2┐ columns and if row m in a tuning precoder column has a non-zero element, so does row m+┌k/2┐.
  • 5. The columns of the tuning precoder 34 for rank r is a subset of the columns of the tuning precoder for rank r+1

With the above in mind, one method herein comprises a method of precoding multi-antenna transmissions 60 from a wireless communication transceiver 10 to another wireless communication transceiver 12. The method includes selecting an overall precoder 36, determining transmission weights for respective ones of two or more transmit antennas 14 according to the selected overall precoder 36, and transmitting weighted signals from the two or more transmit antennas 14 in accordance with the transmission weights. The selected precoder is selected at least in part based on considering precoder information received from the second transceiver 12, which includes indications of precoder selections made by the second transceiver 12, which are intended as precoding recommendations to be considered by the first transceiver 10.

According to the above method, the overall precoder 36 factorizes into a conversion precoder 32 and a tuning precoder 34, wherein the conversion precoder 32 is block diagonal and wherein the tuning precoder 34 has the following properties: all non-zero elements are constant modulus; every column has exactly two non-zero elements; and every row has exactly two non-zero elements; two columns either have non-zero elements in the same two rows or do not have any non-zero elements in the same rows; and two columns having non-zero elements in the same two rows are orthogonal to each other. Further, the conversion precoder 32 has 2┌k/2┐ columns, where k is a non-negative integer, and if row m in a tuning precoder column has a non-zero element, so does row m+┌k/2┐.

Further, in at least one such embodiment, the columns of a tuning precoder 34 for rank r is a subset of the columns of a tuning precoder for rank r+1.

Similarly, another method disclosed herein provides for sending precoding information from a second transceiver 12 to a first transceiver 10 that considers the precoding information in selecting precoders for precoding multi-antenna transmissions 60 to the second transceiver 12.

The method includes the second transceiver 12 selecting an overall precoder 36 that factorizes into a conversion precoder 32 and a tuning precoder 34, or selecting the conversion precoder 32 and the tuning precoder 34 corresponding to a particular overall precoder 36, and sending to the first transceiver 10 as said precoder information an indication of the selected overall precoder 36 or indications of the selected conversion and tuning precoders 32, 34.

For this method, the conversion precoders 32 are each block diagonal and each tuning precoder 34 has the following properties: all non-zero elements are constant modulus; every column has exactly two non-zero elements; and every row has exactly two non-zero elements; two columns either have non-zero elements in the same two rows or do not have any non-zero elements in the same rows; and two columns having non-zero elements in the same two rows are orthogonal to each other. Additionally, according to the method, the conversion precoder 32 has 2┌k/2┐ columns, where k is a non-negative integer, and if row m in a tuning precoder column has a non-zero element, so does row m+┌k/2┐. Still further, in at least one embodiment, the columns of a tuning precoder 34 for rank r is a subset of the columns of a tuning precoder for rank r+1.

Of course, the teachings herein are not limited to the specific, foregoing illustrations. For example, terminology from 3GPP LTE was used in this disclosure to provide a relevant and advantageous context for understanding operations at the transceivers 10 and 12, which were identified in one or more embodiments as being an LTE eNodeB and an LTE UE, respectively. However, the teachings disclosed herein are not limited to these example illustrations and may be advantageously applied to other contexts, such as networks based on WCDMA, WiMax, UMB or GSM.

Further, the transceiver 10 and the transceiver 12 are not necessarily a base station and an item of mobile equipment within a standard cellular network, although the teachings herein have advantages in such a context. Moreover, while certain wireless network examples given herein involve the “downlink” from an eNodeB or other network base station, the teachings presented herein also have applicability to the uplink. More broadly, it will be understood that the teachings herein are limited by the claims and their legal equivalents, rather than by the illustrative examples given herein.

Claims

1. A method in a wireless communication transceiver, wherein another transceiver precodes transmissions to the transceiver based at least in part on the transceiver sending channel state information to the other transceiver that includes precoder information and wherein the method is characterized by:

selecting entries from one or more codebooks as a selected conversion precoder and a selected tuning precoder, or as a selected overall precoder corresponding to a selected conversion precoder and a selected tuning precoder; and

transmitting indications of the selected entries as said precoder information included in the channel state information;

wherein the one or more codebooks include entries comprising n_TQ different conversion precoders, n_T being a number of transmit antenna ports and Q being an integer value, and entries comprising a number of corresponding tuning precoders, or include entries comprising a plurality of overall precoders, with each overall precoder comprising a product of a conversion precoder and a tuning precoder; and

wherein each said conversion precoder out of said n_TQ different entries comprises a block diagonal matrix in which each block comprises a discrete fourier transform (DFT)-based antenna-subgroup precoder that corresponds to a subgroup of n_T transmit antenna ports at the transceiverand provides n_TQ different DFT based beams for the corresponding subgroup, where the n_TQ different conversion precoders, together with one or more of the tuning precoders, correspond to a set of n_TQ different overall precoders, wherein each overall precoder represents a size-n_T DFT-based beam over the n_T transmit antenna ports,;

wherein said one or more codebooks includes conversion and tuning precoders or corresponding overall precoders for multiple transmission ranks,; and

wherein for transmission rank r>2:

the tuning precoder has 2[r/2] rows and r columns, where r is the transmission rank;

all non-zero elements of the tuning precoder are constant modulus;

every column of the tuning precoder has exactly two non-zero elements;

every row of the tuning precoder has exactly two non-zero elements; and

two columns of the tuning precoder having non-zero elements in the same two rows are orthogonal to each other.

2. The method of claim 1, further characterized in that the other transceiver is a base station in a wireless communication network and the transceiver is a user equipment, UE, sending said channel state information to said base station.

3. The method of claim 2, further characterized in that transmitting said indications of the selected entries as the precoder information from the UE to the base station comprises transmitting index values indicating the selected entries within the one or more codebooks.

4. The method of claim 1, further characterized in that said each antenna-subgroup precoder is a matrix block with n_T/2 rows and belongs to a set of n_TQ different DFT-based beams, where Q is an integer equal to or greater than 2, and where each said tuning precoder includes a phase shift element taken from a 2Q Phase Shift Keying (PSK) alphabet and provides at least 2Q relative phase shifts for offsetting beam phases between the antenna-subgroup precoders in a corresponding one of the conversion precoders.

5. The method of claim 1, wherein the columns of a tuning precoder for rank r is a subset of the columns of a tuning precoder for rank r+1.

6. The method of claim 1, further characterized in that each conversion precoder can be written in the form

7. A wireless communication transceiver configured to send channel state information to another wireless communication transceiver that precodes transmissions to the transceiver based at least in part on the channel state information, said transceiver including a receiver for receiving signals from the other transceiver and a transmitter for transmitting signals to the other transceiver, including transmitting signals conveying said channel state information, wherein said transceiver is characterized by:

a memory storing one or more codebooks including entries comprising n_TQ different conversion precoders, n_T being a number of transmit antenna ports, and Q being an integer value, and entries comprising a number of corresponding tuning precoders, or entries comprising a plurality of overall precoders, with each overall precoder comprising a product of a conversion precoder and a tuning precoder, wherein each said conversion precoder out of said n_TQ different entries comprises a block diagonal matrix in which each block comprises a discrete fourier transform (DFT)-based antenna-subgroup precoder that corresponds to a subgroup of n_T transmit antenna ports at the transceiver and provides n_TQ different DFT based beams for the corresponding subgroup, where Q is an integer value and where the n_TQ different conversion precoders, together with one or more of the tuning precoders, correspond to a set of n_TQ different overall precoders, wherein each overall precoder represents a size-n_T DFT-based beam over the n_T transmit antennas ports; and

a precoding feedback generator configured to select entries from the one or more codebooks as a selected conversion precoder and a selected tuning precoder, or as a selected overall precoder corresponding to a selected conversion precoder and a selected tuning precoder;

said precoding feedback generator further configured to transmit, via said transmitter, indications of the selected entries as precoder information included in said channel state information,;

wherein said one or more codebooks includes conversion and tuning precoders or corresponding overall precoders for multiple transmission ranks,; and

wherein for transmission rank r>2:

the tuning precoder has 2[r/2] rows and r columns, where r is the transmission rank;

all of the tuning precoder non-zero elements are constant modulus;

every column of the tuning precoder has exactly two non-zero elements;

every row of the tuning precoder has exactly two non-zero elements; and

two columns of the tuning precoder having non-zero elements in the same two rows are orthogonal to each other.

8. The transceiver of claim 7, further characterized in that the other transceiver is a base station in a wireless communication network and the transceiver is a user equipment, UE, sending said channel state information to said base station.

9. The transceiver of claim 7, further characterized in that each conversion precoder can be written in the form

10. A method of precoding multi-antenna transmissions from a wireless communication transceiver to another wireless communication transceiver, based at least in part on receiving channel state information from the other transceiver that includes precoder information, said method characterized by:

identifying the precoder information by selecting entries from one or more codebooks known at the transceiver responsive to selection indications included in the channel state information; and

precoding a transmission to the other transceiver based at least in part on the precoder information;

wherein the one or more codebooks as known by the transceiver include entries comprising n_TQ different conversion precoders, n_T being a number of transmit antenna ports and Q being an integer value, and entries comprising a number of corresponding tuning precoders, or entries comprising a plurality of overall precoders, with each overall precoder comprising a product of a conversion precoder and a tuning precoder, and wherein each said conversion precoder out of said n_TQ different entries comprises a block diagonal matrix in which each block comprises a discrete fourier transform (DFT)-based antenna-subgroup precoder that corresponds to a subgroup of n_T transmit antenna ports and provides n_TQ different DFT based beams for the corresponding subgroup, wherein the n_TQ different conversion precoders, together with one or more of the tuning precoders, correspond to a set of n_TQ different overall precoders, wherein each overall precoder represents a size- n_T DFT-based beam over the n_T transmit antenna ports,; and

wherein said one or more codebooks includes conversion and tuning precoders or corresponding overall precoders for multiple transmission ranks, and wherein for transmission rank r>2:

the tuning precoder has 2[r/2] rows and r columns, where r is the transmission rank;

all non-zero elements of the tuning precoder are constant modulus;

every column of the tuning precoder has exactly two non-zero elements;

every row of the tuning precoder has exactly two non-zero elements; and

two columns of the tuning precoder having non-zero elements in the same two rows are orthogonal to each other.

11. The method of claim 10, further characterized by performing DFT-based precoding of transmissions from two or more subgroups of the antennas at the transceiver using the antenna-subgroup precoders in one of the conversion precoders, as selected by the transceiver from the one or more codebooks based at least in part on the precoder information.

12. The method of claim 10, further characterized in that each conversion precoder can be written in the form

13. A wireless communication transceiver configured to precode multi-antenna transmissions to another wireless communication transceiver based at least in part on receiving channel state information from the other transceiver, said transceiver including a transmitter and a plurality of antennas for transmitting said multi-antenna transmissions and a receiver for receiving the channel state information, and wherein the transceiver is characterized by:

a memory storing one or more codebooks including entries comprising n_TQ different conversion precoders, n_T being a number of transmit antenna ports and Q being an integer value, and entries comprising a number of corresponding tuning precoders, or entries comprising a plurality of overall precoders, with each overall precoder comprising a product of a conversion precoder and a tuning precoder, wherein each said conversion precoder out of said n_TQ different entries comprises a block diagonal matrix in which each block comprises a discrete fourier transform (DFT)-based antenna-subgroup precoder that corresponds to a subgroup of n_T transmit antenna ports and provides n_TQ different DFT based beams for the corresponding subgroup, where the n_TQ different conversion precoders together with one or more of the tuning precoders correspond to a set of n_TQ different overall precoders, wherein each overall precoder represents a size- n_T DFT-based beam over the n_T transmit antenna ports;

a feedback processor configured to identify precoder information from the other receiver based on using selection indications included in the channel state information to identify from the one or more codebooks selected conversion and tuning precoders or a selected overall precoder corresponding to selected conversion and tuning precoders; and

a precoding controller and associated precoding circuit configured to precode the transmission to the other transceiver, based at least in part on the precoder information,;

wherein said one or more codebooks includes conversion and tuning precoders or corresponding overall precoders for multiple transmission ranks,; and

wherein for transmission rank r>2:

the tuning precoder has 2[r/2] rows and r columns, where r is the transmission rank;

all non-zero elements of the tuning precoder are constant modulus;

every column of the tuning precoder has exactly two non-zero elements;

every row of the tuning precoder has exactly two non-zero elements,; and

two columns of the tuning precoder having non-zero elements in the same two rows are orthogonal to each other.

14. The transceiver of claim 13, further characterized in that the precoding controller and associated precoding circuit are configured to precode the transmission to the other transceiver by performing DFT-based precoding of transmissions from two or more subgroups of the antennas using antenna-subgroup precoders in the conversion or overall precoder selected by the transceiver from the one or more codebooks, where said selection by the transceiver is based at least in part on the precoder information.

15. The transceiver of claim 13, further characterized in that each conversion precoder can be written in the form